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### Monte Carlo Event Generators

Lecture 2: Parton Showers and Perturbative QCD

Peter Richardson

IPPP, Durham University

CTEQ School July 06

Plan

- Lecture 1:Introduction
- Basic principles of event generation
- Monte Carlo integration techniques
- Matrix Elements
- Lecture 2:Parton Showers
- Parton Shower Approach
- Recent advances, CKKW and [email protected]
- Lecture 3:Hadronization and Underlying Event
- Hadronization Models
- Underlying Event Modelling

CTEQ School July 06

Lecture 2

Today we will cover

- The Standard Parton Shower approach
- Final-state showers
- Initial-state showers.
- Merging Matrix Elements with Parton Showers
- Traditional Approach
- CKKW
- [email protected]
- Where Next?

CTEQ School July 06

e+e- Annihilation to Jets

Singular

- In order to consider the basic idea of the parton shower let’s start with the cross section for e+e- annihilation into three jets.
- The cross section can be written as
- where x1and x2 are the momentum fractions of the quark and antiquark
- Singular as x1g1 or x2g1.
- Energy Conservation requires

1

0

1

CTEQ School July 06

e+e- Annihilation to Jets

- So the matrix element is singular as x1g1, x2g1 or both.
- What does this mean physically?
- So x2g1 means that 1 and 3 are collinear.
- Also x1+x2+x3=2 means that x1g1 and x2g1 implies xgg0, soft

CTEQ School July 06

QCD Radiation

- It is impossible to calculate and integratethe matrix elements for large numbers of partons.
- Instead we treat the regions where the emission of QCD radiation is enhanced.
- This is soft and collinear radiation.

CTEQ School July 06

Collinear Singularities

- In the collinear limit the cross section for a process factorizes
- Pji(z) is the DGLAP splitting function.
- The splitting function only depends on the spin and flavours of the particles

CTEQ School July 06

Splitting Functions

CTEQ School July 06

Collinear Singularities

- This expression is singular as qg0.
- What is a parton? (or what is the difference between a collinear pair and a parton)
- Introduce a resolution criterion, e.g.
- Combine the virtual corrections and unresolvable emission

Resolvable Emission

Finite

Unresolvable Emission

Finite

- Unitarity: Unresolved + Resolved =1

CTEQ School July 06

Monte Carlo Procedure

- Using this approach we can exponentiate the real emission piece.
- This gives the Sudakov form factorwhich is the probability of evolving between two scales and emitting no resolvable radiation.
- More strictly it is the probability of evolving from a high scale to the cut-off with no resolvable emission.

CTEQ School July 06

Start with an isotope

Work out when it decays by generating a random number and solving

where t is its lifetime

Generate another random number and use the branching ratios to find the decay mode.

Generate the decay using the masses of the decay products and phase space.

Repeat the process for any unstable decay products.

This algorithm is actually used in Monte Carlo event generators to simulate particle decays.

Start with a parton at a high virtuality, Q, typical of the hard collision.

Work out the scale of the next branching by generating a random number and solving

where q is the scale of the next branching

If there’s no solution for q bigger than the cut-off stop.

Otherwise workout the type of branching.

Generate the momenta of the decay products using the splitting functions.

Repeat the process for the partons produced in the branching.

Numerical ProcedureParton Shower

Radioactive Decay

CTEQ School July 06

Monte Carlo Procedure

- The key difference between the different Monte Carlo simulations is in the choice of the evolution variable.
- Evolution Scale
- Virtuality, q2
- Transverse Momentum, kT.
- Angle, q.
- ….
- Energy fraction, z
- Energy fraction
- Light-cone momentum fraction
- ….
- All are the same in the collinear limit.

CTEQ School July 06

Soft Emission

- We have only considered collinear emission. What about soft emission?
- In the soft limit the matrix element factorizes but at the amplitude level.
- Soft gluons come from all over the event.
- There is quantum interference between them.
- Does this spoil the parton shower picture?

CTEQ School July 06

Angular Ordering

Colour Flow

- There is a remarkable result that if we take the large number of colours limit much of the interference is destructive.
- In particular if we consider the colour flow in an event.
- QCD radiation only occurs in a cone up to the direction of the colour partner.
- The best choice of evolution variable is therefore an angular one.

Emitter

Colour Partner

CTEQ School July 06

Colour Coherence

- Angular Ordering and ColourCoherence are often used interchangeably in talks etc..
- However there is a difference.
- Colour Coherence is the phenomena that a soft gluon can’t resolve a small angle pair of particles and so only sees the colour charge of the pair.
- Angular Ordering is a way of implementing colour coherence in parton shower simulations.

CTEQ School July 06

Running Coupling

- It is often said that Monte Carlo event generators are leading-log.
- However they include many effects beyond leading log, e.g.
- Momentum Conservation
- Running Coupling Effects
- Effect of summing higher orders is absorbed by replacing as with as(kT2).
- Gives more soft gluons, but must avoid the Landau pole which makes the cut-off a physical parameter.

CTEQ School July 06

Initial-State Radiation

- In principle this is similar to final-state radiation.
- However in practice there is a complication
- For final-state radiation
- One end of the evolution fixed, the scale of the hard collision.
- For initial-state radiation
- Both ends of the evolution fixed, the hard collision and the incoming hadron
- Use a different approach based on the evolution equations.

CTEQ School July 06

Initial-State Radiation

- There are two options for the initial-state shower
- Forward Evolution
- Start at the hadron with the distribution of partons given by the PDF.
- Use the parton shower to evolve to the hard collision.
- Reproduces the PDF by a Monte Carlo procedure.
- Unlikely to give an interesting event at the end, so highly inefficient.
- Backward Evolution
- Start at thehard collision and evolve backwards to the proton guided by the PDF.
- Much more efficient in practice.

CTEQ School July 06

Initial-State Radiation

- The evolution equation for the PDF can be written as
- Or
- This can be written as a Sudakov form-factor for evolving backwards in time, i.e from the hard collision at high Q2 to lower with

CTEQ School July 06

The Colour Dipole Model

- The standard parton shower approach starts from the collinear limit and makes changes to include soft gluon coherence.
- The Colour Dipole Model starts from the soft limit.
- Emission of soft gluons from the colour-anticolour dipole is universal.
- After emitting a gluon, the colour dipole splits into two new dipoles

i

CTEQ School July 06

Parton Shower

- ISAJET uses the original parton shower algorithm which only resums collinear logarithms.
- PYTHIA uses the collinear algorithm with an angular veto to try to reproduce the effect of the angular ordered shower.
- HERWIG uses the angular ordered parton shower algorithm which resums both soft and collinear singularities.
- SHERPA uses the PYTHIA algorithm.
- ARIADNE uses the colour dipole model.

CTEQ School July 06

LEP Event Shapes

CTEQ School July 06

Hadron Collisions

- The hard scattering sets up the initial conditions for the parton shower.
- Colour coherence is important here too.
- Each parton can only emit in a cone stretching to its colour partner.
- Essential to fit the Tevatron data.

CTEQ School July 06

Hadron Collisions

- Distributions of the pseudorapidity of the third jet.
- Only described by
- HERWIG which has complete treatment of colour coherence.
- PYTHIA+ has partial
- PRD50, 5562, CDF (1994)

CTEQ School July 06

Recent Progress

- In the parton shower per se there have been two recent advances.
- New Herwig++ shower
- Based on massive splitting functions.
- Better treatment of radiation from heavy quarks.
- More Lorentz invariant.
- New PYTHIA pT ordered shower
- Order shower in pT, should be coherent.
- Easier to include new underlying event models.
- Easier to match to matrix elements

CTEQ School July 06

Herwig++ for tgbW+g

- Based on the formalism of S. Gieseke, P. Stephens and B.R. Webber,JHEP 0312:045,2003.
- Improvement on the previous FORTRAN version.

CTEQ School July 06

PYTHIA pT ordered Shower

CTEQ School July 06

Hard Jet Radiation

- The parton shower is designed to simulate soft and collinear radiation.
- While this is the bulk of the emission we are often interested in the radiation of a hard jet.
- This is not something the parton shower should be able to do, although it often does better than we except.
- If you are looking at hard radiation HERWIG/PYTHIA will often get it wrong.

CTEQ School July 06

Hard Jet Radiation

- Given this failure of the approximations this is an obvious area to make improvements in the shower and has a long history.
- You will often here this called
- Matrix Element matching.
- Matrix Element corrections.
- Merging matrix elements and parton shower
- [email protected]
- I will discuss all of these and where the different ideas are useful.

CTEQ School July 06

Hard Jet Radiation: General Idea

- Parton Shower (PS) simulations use the soft/collinear approximation:
- Good for simulating the internal structure of a jet;
- Can’t produce high pT jets.
- Matrix Elements (ME) compute the exact result at fixed order:
- Good for simulating a few high pT jets;
- Can’t give the structure of a jet.
- We want to use both in a consistent way, i.e.
- ME gives hard emission
- PS gives soft/collinear emission
- Smooth matching between the two.
- No double counting of radiation.

CTEQ School July 06

Matching Matrix Elements and Parton Shower

Parton Shower

- The oldest approaches are usually called matching matrix elements and parton showers or the matrix element correction.
- Slightly different for HERWIG and PYTHIA.
- In HERWIG

HERWIG phase space for Drell-Yan

Dead Zone

- Use the leading order matrix element to fill the dead zone.
- Correct the parton shower to get the leading order matrix element in the already filled region.
- PYTHIA fills the full phase spaceso only the second step is needed.

CTEQ School July 06

Matrix Element Corrections

Z qT distribution from CDF

W qT distribution from D0

G. Corcella and M. Seymour, Nucl.Phys.B565:227-244,2000.

CTEQ School July 06

Matrix Element Corrections

- There was a lot of work for both HERWIG and PYTHIA. The corrections for
- e+e- to hadrons
- DIS
- Drell-Yan
- Top Decay
- Higgs Production

were included.

- There are problems with this
- Only the hardest emission was correctly described
- The leading order normalization was retained.

CTEQ School July 06

Recent Progress

- In the last few years there has been a lot of work addressing both of these problems.
- Two types of approach have emerged
- NLO Simulation
- NLO normalization of the cross section
- Gets the hardest emission correct
- Multi-Jet Leading Order
- Still leading order.
- Gets many hard emissions correct.

CTEQ School July 06

NLO Simulation

- There has been a lot of work on NLO Monte Carlo simulations.
- Only the [email protected] approach of Frixione, Nason and Webber has been shown to work in practice.
- Although an alternative approach by Nason looks promising and a paper with results for Z pairs appeared last week.

CTEQ School July 06

[email protected]

- [email protected] was designed to have the following features.
- The output is a set of fully exclusive events.
- The total rate is accurate to NLO
- NLO results for observables are recovered when expanded in as.
- Hard emissions are treated as in NLO calculations.
- Soft/Collinear emission are treated as in the parton shower.
- The matching between hard emission and the parton shower is smooth.
- MC hadronization models are used.

CTEQ School July 06

Basic Idea

- The basic idea of [email protected] is
- Work out the shower approximation for the real emission.
- Subtract it from the real emission from
- Add it to the virtual piece.
- This cancels the singularities and avoids double counting.
- It’s a lot more complicated than it sounds.

CTEQ School July 06

Toy Model

- I will start with Bryan Webber’s toy model to explain [email protected] to discuss the key features of NLO, MC and the matching.
- Consider a system which can radiate photons with energywith energy with

where is the energy of the system before radiation.

- After radiation the energy of the system
- Further radiation is possible but photons don’t radiate.

CTEQ School July 06

Toy Model

- Calculating an observable at NLO gives

where the Born, Virtual and Real contributions are

a is the coupling constant and

CTEQ School July 06

Toy Model

- In a subtraction method the real contribution is written as
- The second integral is finite so we can set
- The NLO prediction is therefore

CTEQ School July 06

Toy Monte Carlo

- In a MC treatment the system can emit many photons with the probability controlled by the Sudakov form factor, defined here as

where is a monotonic function which has

- is the probability that no photon can be emitted with energy such that .

CTEQ School July 06

Toy [email protected]

- We want to interface NLO to MC. Naïve first try
- start MC with 0 real emissions:
- start MC with 1 real emission at x:
- So that the overall generating functional is
- This is wrong because MC with no emissions will generate emission with NLO distribution

CTEQ School July 06

Toy [email protected]

- We must subtract this from the second term
- This prescription has many good features:
- The added and subtracted terms are equal to
- The coefficients of and are separately finite.
- The resummation of large logs is the same as for the Monte Carlo renormalized to the correct NLO cross section.

However some events may have negative weight.

CTEQ School July 06

Toy [email protected] Observables

- As an example of an “exclusive” observable consider the energy y of the hardest photon in each event.
- As an “inclusive” observable consider the fully inclusive distributions of photon energies, z
- Toy model results shown are for

CTEQ School July 06

Toy [email protected] Observables

CTEQ School July 06

Real QCD

- For normal QCD the principle is the same we subtract the shower approximation to the real emission and add it to the virtual piece.
- This cancels the singularities and avoids double counting.
- It’s a lot more complicated.

CTEQ School July 06

Problems

- For each new process the shower approximation must be worked out, which is often complicated.
- While the general approach works for any shower it has to be worked out for a specific case.
- So for [email protected] only works with the HERWIG shower algorithm.
- It could be worked out for PYTHIA or Herwig++ but this remains to be done.

CTEQ School July 06

HERWIG

NLO

W+W- ObservablesPT of W+W-

Dj of W+W-

[email protected] gives the correct high PT result and soft resummation.

S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

CTEQ School July 06

HERWIG

NLO

W+W- Jet ObservablesS. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

CTEQ School July 06

HERWIG

NLO

Top ProductionS. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

CTEQ School July 06

HERWIG

NLO

Top Production at the LHCS. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

CTEQ School July 06

B Production at the Tevatron

S. Frixione, P. Nason and B.R. Webber, JHEP 0308(2003) 007, hep-ph/0305252.

CTEQ School July 06

Higgs Production at LHC

S. Frixione and B.R. Webber JHEP 0206(2002) 029, hep-ph/0204244, hep-ph/0309186

CTEQ School July 06

NLO Simulation

- So far [email protected] is the only implementation of a NLO Monte Carlo simulation.
- Recently there have been some ideas by Paulo Nason JHEP 0411:040,2004 and recent results.
- In this approach there are no negative weights but more terms would be exponentiated beyond leading log.

CTEQ School July 06

Multi-Jet Leading Order

- While the NLO approach is good for one hard additional jet and the overall normalization it cannot be used to give many jets.
- Therefore to simulate these processes use matching at leading order to get many hard emissions correct.
- I will briefly review the general idea behind this approach and then show some results.

CTEQ School July 06

CKKW Procedure

- Catani, Krauss, Kuhn and Webber JHEP 0111:063,2001.
- In order to match the ME and PS we need to separate the phase space:
- One region contains the soft/collinear region and is filled by the PS;
- The other is filled by the matrix element.
- In these approaches the phase space is separated using in kT-type jet algorithm.

CTEQ School July 06

Durham Jet Algorithm

- For all final-state particles compute the resolution variables
- The smallest of these is selected. If is the smallest the two particles are merged. If is the smallest the particle is merged with the beam.
- This procedure is repeated until the minimum value is above some stopping parameter .
- The remaining particles and pseudo-particles are then the hard jets.

CTEQ School July 06

CKKW Procedure

- Radiation above a cut-off value of the jet measure is simulated by the matrix element and radiation below the cut-off by the parton shower.
- Select the jet multiplicity with probability

where is the n-jet matrix element evaluated at resolution using as the scale for the PDFs and aS, n is the number of jets

- Distribute the jet momenta according the ME.

CTEQ School July 06

CKKW Procedure

- Cluster the partons to determine the values at which 1,2,..n-jets are resolved. These give the nodal scales for a tree diagram.
- Apply a coupling constant reweighting.

CTEQ School July 06

CKKW Procedure

- Reweight the lines by a Sudakov factor
- Accept the configuration if the product of the aS and Sudakov weight is less than otherwise return to step 1.

CTEQ School July 06

CKKW Procedure

- Generate the parton shower from the event starting the evolution of each parton at the scale at which it was created and vetoing emission above the scale .

CTEQ School July 06

CKKW Procedure

- Although this procedure ensures smooth matching at the NLL log level are still choices to be made:
- Exact definition of the Sudakov form factors.
- Scales in the strong coupling and aS.
- Treatment of the highest Multiplicity matrix element.
- Choice of the kT algorithm.
- In practice the problem is understanding what the shower is doing and treating the matrix element in the same way.

CTEQ School July 06

e+e- Results from SHERPA

CTEQ School July 06

What Should I use?

- Hopefully this lecture will help you decide which of the many different tools is most suitable for a given analysis.
- Only soft jets relative to hard scale MC
- Only one hard jet [email protected] or old style ME correction
- Many hard jets CKKW.
- The most important thing is to think first before running the simulation.

CTEQ School July 06

Summary

- In this afternoon’s lecture we have looked at
- The basic parton shower algorithm
- Colour Coherence
- Backward Evolution
- Next-to-leading Order simulations
- Matrix Element matching
- On Thursday we will go on and look at the non-perturbative parts of the simulation.

CTEQ School July 06

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